1. Clickers
Bellwork
Translate the following statement into a conditional statement
Angles measuring less than 90o are acute angles
Write the converse, inverse & contrapositive of the conditional statement you just wrote
Determine the validity of each of the four statements you’ve created
Can the statement “legally” be changed into a bi-conditional statement

2. Bellwork Solution

3. Analyze Deductive Reasoning
Section 2.3

4. The Concept
Deductive reasoning is one of other major components in the formation of logical arguments
Today we’re going to see how deductive reasoning works and how we can apply it to mathematics

5. Definition Deductive Reasoning
Process of making logical statements by using facts, definitions, accepted properties and the laws of logic.
Axis of symmetry
Vertex

6. What’s the difference Inductive Reasoning Deductive Reasoning
Uses specific examples to make conjectures for the larger case
Does not use any information outside of direct observation
Deductive Reasoning
Uses accepted facts and laws to make conjectures without the use of specific examples
For example
All the other cars around me are going 55 mph, I should too.
The sign says the Speed Limit is 55 mph. Because it is an official sign placed by the proper authorities, I should follow it
Other examples?
Axis of symmetry
Vertex

7. Reasoning from a graph
Is the statement a product of inductive or deductive reasoning
A 120 lb person uses twice as many calories when cycling 4 hours than when cycling 2 hours
A. Inductive
B. Deductive
A 120 lb person uses fewer calories the longer he or she rides
A. Inductive
B. Deductive

8. Law of Detachment Law of Detachment
If the hypothesis of a conditional statement is true, then the conclusion must also be true
Basic assumption of truth based on credible conditional statements
For example
If two angles are right angles, then they are congruent. ABC & DEF are both right angles.
If you are a UA student, then you are wearing a polo shirt. You are a UA student.
Common parenting tool
If you eat your chicken, then you’ll get dessert. You ate your chicken.
Axis of symmetry
Vertex

9. Example If x>12, then –x<-12. The value of x is 15.
Axis of symmetry
Vertex

10. Law of Syllogism Law of Syllogism Often associated with Socrates
Process of stringing together two true conditional statements to prove a conditional statement
If p, then q is true, and if q, then r is true. Then, if p then r is true
Often associated with Socrates
Used this method to both prove points and to disprove statements of others
For example
Xenophon, a tax collector, states that he is a pious person .
1: If you are a pious person, then you do not covet money
2. If you do not covet money, then you cannot be a tax collector
3. If you are a pious person, then you cannot be a tax collector
Axis of symmetry
Vertex

11. Examples
What two conditional statements can we use to prove the following conditional statement
If two angles are supplementary, then together they form a straight angle
If two angles are supplementary, then form a 180o angle
If an angle measures 1800, then it is a straight angle

12. Tandem
The reason that both inductive a deductive reasoning are learned is so that conjectures can be made via experience and then proved via the laws of logic
On your first day of school, you notice that everyone stays seated in a class until the teacher dismisses them
Conjecture:
In order to leave class at this school, you have to be dismissed by the teacher
Proof:
The teacher dictates how the students act
The students act in a certain way when they leave a class
The teacher determines how students leave a class

13. Examples
What kind of statement can we make based on the following conditional using the law of detachment
If the walls are painted red, then the students will get angry
The walls just got painted red
The students don’t like red
The students will get angry
The painters really like red
Red is a fun color
Vertex

14. Examples
What kind of statement can we make based on the following conditionals using the Law of Syllogism
If you aren’t wearing a belt, then you’re out of dress code
If you’re out of dress code, then you have to visit the office
If you out of dress code, then you’re not wearing a belt
If you visit the office, then you aren’t wearing a belt
If you visit the office, then you’re out of dress code
If you aren’t wearing a belt, then you have to visit the office
Vertex

15. Examples
What kind of statement can we make based on the following conditionals using the Law of Syllogism
If you don’t eat breakfast, then you’re metabolism slows down
If you’re brain doesn’t function optimally, then you’re metabolism is slow
If you don’t eat breakfast, then you’re brain doesn’t function optimally
If you’re metabolism is slow, then you didn’t eat breakfast
If you’re brain doesn’t function optimally, then you didn’t eat breakfast
If you’re brain is slow, then your metabolism isn’t functioning optimally
Vertex

16. Homework
2.3
1-3, 4-14 even, 16-19, 21-28

17. Examples
What kind of statement can we make based on the following conditionals using the Law of Syllogism
If you drive to school, then you park in the parking lot.
If you have a parking permit, then you can park in the lot.
Vertex

18. Most Important Points Deductive Reasoning Law of Detachment
Law of Syllogism